12
12
12
12
12
12
12
12
12 T
TWELF
WELF
1
2
12 12
12
12
Do Twelfths
Terminate or Repeat?
w
Rebecca Ambrose and Erica Burnison
When we find the decimal equivalent
of a fraction with 12 in the denominator, will it terminate or repeat? This
question came from a seventh grader
in author Erica Burnison’s class as the
student was pondering a poster generated by one of her classmates. Not
only did we find the question intriguing, but it also affirmed our belief in
the power of tailoring instruction to
students’ interests and needs.
As math educators, trying to figure
out how to implement the content
230
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
●
Vol. 21, No. 4, November 2015
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and Standards for Mathematical Practice (SMP) found in the Common
Core State Standards for Mathematics in a student-centered fashion, we
were thrilled by this moment. This
question from a student also demonstrated that students can engage in the
SMP while working in collaborative
groups.
In this article, we will describe the
activities that gave rise to this studentgenerated question. We will also
provide our collaborative reflection of
12
12
12
3/12 1/4.25
6/12 1/2.5
8/12 2/3.666666666
12
12
FTHS
12
12
12 12
While seventh-grade students group fractions, they learn more about
decimals, in particular, and division, in general.
the activities because we find that collaboration deepens our understanding
of pedagogy as well as mathematics.
We hope that our account will inspire
others to work together.
We work in a university-school
partnership, Innovations in STEM
Teaching, Achievement, and Research
(I-STAR). Burnison is also involved
with the BetterLesson Master Teacher
Project, where she documented her
teaching in 2013–14. These two projects gave her access to videorecordings
of her students’ conversations. Viewing videotapes of group discussions
after class enabled her to eavesdrop
on her students and use their ideas to
tailor subsequent lessons to address
their needs as well as meet the content
standards.
INITIAL LESSON SET UP
The focus of the lessons we describe
explores a grade 7 number standard in
the Common Core: “Convert a rational
number to a decimal using long diviVol. 21, No. 4, November 2015
●
sion; know that the decimal form of
a rational number terminates in 0s or
eventually repeats” (CCSSI 2010, 7.NS
2d; p. 49). Because Burnison’s seventhgrade students typically exhibited a
negative attitude about their math
abilities and tended to be fearful of
anything related to rational numbers,
she anticipated knowledge gaps. She
wanted to challenge her students’
misconceptions, so she began by asking
them to convert the fraction 5/8 to a
decimal. When they asked, “Does
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
231
Fig. 1 Students were asked to produce equivalent fraction and decimal pairs from the
ERICA BURNISON
set below, then sort them in any way they pleased.
JUPITERIMAGES/THINKSTOCK
the 5 or the 8 go in the house?” and
did not know whether 0.625 or 1.6
was a reasonable answer, she knew
that they needed to develop better
rational number sense. Difficulties
with decimals are typical for many
students, as identified in the research
literature (Moloney and Stacey 1997).
Burnison hoped to change the
students’ mind-set that fractions
and decimals were the
enemy by providing them with an
opportunity to explore rational
numbers in a nonthreatening way.
Armed with the information about
students’ misconceptions, she created
a set of fractions for small cooperative
groups to use in a sorting task. She
strategically chose fractions in the
hopes that students would observe
patterns that might spark conjectures
and questions (see fig. 1). She wanted
to hear her students talk about “What
if ” and “Why” rather than “How to”
as they developed a better understanding of the connection between
fractions and decimals. She also
hoped the task would help them confront some of their misconceptions.
To complete the task, students
converted each fraction into a decimal
and then used a calculator to doublecheck their work. Once they had the
equivalent fraction and decimal pairs,
students were asked to sort the fractions into sets in any way that made
sense to them. Burnison did not care
how they chose to sort them; she
wanted them to look beyond the procedures that seemed to defeat them
and instead experience the excitement
of discovery that she hoped would
emerge from noticing patterns.
Students represented their sorting discoveries on posters (see fig. 2).
Some groups sorted the fractions according to same denominators; others
separated repeating and terminating
decimals; and others distinguished
between fractions that were more or
less than one whole. The variety of
responses provided fodder for good
discussions in future lessons, so the
effort to tailor instruction to student
needs could continue. At this point
in the unit, students had experienced
using division to convert fractions to
decimals. Their engagement with the
SMP had only just begun.
ONE GROUP’S INTERACTION
Burnison examined the posters
and decided that she would provide prompts to engage students in
examining one another’s work and
[The teacher] chose specific fractions
because she wanted to hear “What if”
and “Why” rather than “How to.”
thinking about what classmates had
completed. The prompts helped
students use the organizational patterns created by their classmates to
explore the Common Core’s content
standard.
The prompt that went with the
poster in figure 2 was this:
Is there anything you notice
about the fractions that tells
you in which place value the
decimal ends or how the
decimal ends?
These students had grouped all the
fractions with the same denominator.
In viewing the arrangements,
Sam’s, Elena’s, and Juana’s attention
was drawn to the fractions with
8 in the denominator. They claimed
that “when you divide by 8, it always
ends in the thousandths place and it
always ends in a 5.” At this point, the
trio had already noticed that fractions
with 8 in the denominator convert to
decimals that terminate with a 5 in
the thousandths place. Sam continued to look for additional examples
to support their claim, but, finding
none, moved on to the next pattern.
In retrospect, we realized that an
opportunity was missed to suggest to
the students that they generate their
own data by providing more examples (i.e., 2/8 or 3/8) and testing their
Fig. 2 It was hoped that students would find patterns while they were sorting fractions.
claim; however, we were happy that
at least Sam realized that more data
would strengthen his argument.
Elena then asked the group,
“What about dividing by 2, 4, or 6?”
They scanned the poster again and
noticed that dividing by 2 also results
in a decimal that “ends in a 5.” Elena
then pointed to 5/2 = 2.5, 9/2 = 4.5,
and 1/2 = 0.5 to illustrate her point.
Sam repeated the conjecture that
“when you divide by 2, when the denominator is 2, it ends in a 5.” These
students had recognized another
pattern in the relationship between
the denominator and the terminating
decimal equivalent, but had not asked
why it occurred. More observations
ensued.
Elena: Well, dividing by 2 and 4, it
will end in a 5, too.
Sam: [Continued focusing on
fractions with 2 in the denominator] These are all tenths. [Then
he pointed to 3/4 = 0.75.] But
this one ends in the hundredths,
and the eighths ends in the
thousandths. They all have 5.
Juana: Not all of them.
Sam: [Revising his statement]
Yeah, the higher the number
it is, the higher the denominator;
well, it has to be 2, 4, 8.
Elena: Wait, are you saying all of
them? All of them end in 5?
Sam: Only 2, 4, and 8 all end in 5,
but 8s always in the. . .
Juana: Thousandths.
ERICA BURNISON
Sam and Juana then stated the rest of
their claim together as they pointed
to examples on the poster, “Twos
always in the tenths, and fourths are
in the hundredths.”
Elena was particularly persistent
in her critique by asking which
fractions Sam was talking about,
causing Sam to engage in SMP 6,
“Attend to precision,” as he specified
which fractions ended in a 5.
Vol. 21, No. 4, November 2015
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MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
233
ANOTHER GROUP’S
INTERACTION
Another group had created a poster
that separated the fractions into
categories of repeating and nonrepeating decimals. Burnison decided to use
this poster, illustrated in figure 3, to
help students explore the patterns in
repeating decimals and the division
that produced them.
This group noticed that all the denominators for the repeating decimals
were multiples of 3 (i.e., 3, 6, and 9).
When Adam listed the denominators out loud, he did not stop at the
examples provided on the poster but
extended his pattern to denominators
of 12. Isabella immediately questioned
his assertion and pointed out that
12 is a multiple of both 3 and 4. She
wondered if, when converting fractions to decimals, a denominator of 12
would act more like 3 and repeat or
more like 4 and terminate ending in
5. Burnison was delighted that Adam
extended the pattern and that Isabella
critiqued his reasoning.
Both groups of students had
noticed and expressed regularity
(SMP 8) and had identified and used
the structure of the number system
(SMP 7). The first group noticed
two regularities: The fractions whose
equivalent decimal ended in 5 had deFig. 3 This listing demonstrated why
denominators with powers of 2 convert
to decimals that end in a 5.
1 1 5 5
= • =
= 0.5
2 2 5 10
1 1
1 52
52
25
= 2 = 2• 2 = 2 =
= 0.25
4 2
2 5
10
100
1 1
1 53
53
125
= 3 = 3• 3 = 3 =
= 0.125
8 2
2 5
10
1000
234
nominators of 2, 4, and 8 and, within
that group of fractions, the number
of digits after the decimal point
increased by 1 as the denominator
doubled. The second group noticed
that all the repeating decimals had
fraction equivalents with a multiple of
3 in the denominator. Moreover, they
were extending the pattern to fractions with a denominator of 12 and
conjecturing about how its decimal
equivalents would behave. By articulating their observations to one
another, they were working toward
SMP 3, “Construct a viable argument
and critique the reasoning of others.”
Although their arguments could use
some fine-tuning, we were excited to
see them engaging independently in
the Common Core’s Standards for
Mathematical Practice. They understood Burnison’s expectation that they
make sense of things for themselves
and share their ideas with classmates,
and their interactions indicated that
they were wholeheartedly participating in the activity. Their discussions
provided Burnison with a window
into their thinking and another opportunity to tailor her instruction to
their needs.
STUDENTS’ QUESTIONS
The next day, the class was told about
the question that Adam and Isabella
had brought up, “What happens when
fractions with 12 in the denominator
are turned into decimals?” and were
invited to figure it out for themselves.
Burnison designed an activity sheet
to help gather data, listing all the
twelfths from 1/12 through 12/12.
The students were eager to get started
trying to answer their classmates’
question. After simplifying the fractions, most students used the division
algorithm to convert the fractions
to decimals. Burnison was pleased
to see that students were using their
knowledge of the size of the fraction
to decide how to set up the division, a
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
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Vol. 21, No. 4, November 2015
procedure that initially had thwarted
them.
In one group, Marianna suggested
that all the fractions that could be
simplified would be converted to repeating decimals. However, Francisco
and Becka pointed to the fractions
3/12, 6/12, and 8/12 as counterexamples to Marianna’s claim. The
group persevered and continued to
look for patterns in the data. Like
most of the groups, these students
concluded that the fractions, once
written in lowest terms, did hold the
key to deciding what denominators
of 12 would do. Those that simplified
to a denominator of 3 would convert
to repeating decimals, and those that
simplified to denominators of 2 or 4
would terminate with a 5 in the tenths
and hundredths place, respectively.
Another group made a slightly different observation, asserting that if a
factor of 4 was removed when simplifying, the resulting decimal would repeat. Alternatively, if a factor of 3 was
removed, the fraction would convert
to a terminating decimal. Students
were discovering relationships among
fractions, decimals, and division and
engaging in the Standards for Mathematical Practice.
Burnison was so focused on accomplishing this goal that she missed
an opportunity to dig deeper into the
math when Miguel shared his thinking. While completing the activity
sheet, Miguel realized that he did not
need to use long division to convert
fractions to decimals, but instead
could think about equivalent fractions
with powers of 10 in the denominator. He did not articulate this pattern,
but noted that 3/4 = 75/100 = 0.75. In
this case, Miguel had found a useful
structure in the numbers (SMP 7).
He stepped back and shifted perspective to notice the equivalent fractions
embedded in the process of converting fractions to decimals. Burnison
did not immediately appreciate this
Students concluded that the fractions, once
written in lowest terms, did hold the key to
deciding what denominators of 12 would do.
ERICA BURNISON
underlying big idea in the midst of
instruction.
Sometimes it is difficult to recognize these opportunities during class
and make adjustments in the moment.
After we, the authors, collaborated,
the important mathematical idea
that Miguel was approaching became
clear. We realized that the students
were looking for patterns but were
not asking what was causing them.
In particular, they did not provide a
justification for why fractions with
a power of 2 in the denominator
(2, 4, 8) would terminate in a 5 when
converted to a decimal.
Although we discussed the math
in figure 3, we are not sure if sev-
enth graders could have generated
this analysis using exponents. Even
so, Burnison appreciated the discussion because she realized that thinking about equivalent fractions in
relation to converting fractions to
decimals would allow her to extend
her students’ thinking should the opportunity arise. She recognized that
emphasizing decimals as a “special
kind of fraction with a power of 10 in
the denominator” would help her students understand why some fractions
terminated and why others did not.
In retrospect, she wished that she had
engaged the whole class in thinking
about Miguel’s approach so she could
see how far they could go with it (see
Kreith 2014 for a more extensive
discussion of fig. 3).
FROM DIVISION PHOBIA TO
UNDERSTANDING
Burnison’s students had initially been
division-phobic and were lacking
number sense with respect to fractions and decimals. Some teachers
might avoid exploring the relationship between fractions and decimals
until after students had mastered long
division and after students understood
fractions better. Instead, Burnison
had students explore terminating
and repeating decimals; in so doing,
it bolstered their abilities in division
and their understanding of fractions
simultaneously. Although Burnison
addressed several practice standards,
she continually adjusted instruction to
take advantage of her students’ contributions. She used the posters they
produced as a resource and presented
a student-generated question for all
to investigate. She made it clear that
Students
organized
their fractions
in many
different
ways.
Vol. 21, No. 4, November 2015
●
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
235
she valued their work and found their
ideas worthy of consideration. In
response, students engaged in several
SMP, actively looked for patterns,
searched for structure, articulated their
thinking, and attended to precision.
She was able to tailor instruction
to her students’ needs and input, meet
the content standard, and facilitate the
math practices because she had faith
in her students’ curiosity and in their
intelligence. In addition, she explicitly valued their thinking. She had
nurtured a classroom culture in which
students felt comfortable sharing their
thinking and questioning each other.
Students’ conversations fueled their
thinking, and together they came
up with interesting ideas to pursue.
This series of lessons felt like happy
accidents that followed from students’
ideas and questions. But really they
were a result of lessons purposely
designed to elicit student thinking which, in turn, provided further
avenues for exploring math.
REFERENCES
Common Core State Standards Initiative (CCSSI). 2010. Common Core
State Standards for Mathematics.
Washington, DC: National Governors
Association Center for Best Practices
and the Council of Chief State School
Officers. http://www.corestandards
.org/wp-content/uploads/Math_
Standards.pdf
Kreith, Kurt. 2014. “Fractions, Decimals,
and the Common Core.” Journal of
Mathematics Education at Teachers
College 5 (Spring-Summer): 19–25.
Moloney, Kevin, and Kaye Stacey. 1997.
“Changes with Age in Students’
Conceptions of Decimal Notation.”
Mathematics Education Research
Journal 9 (1): 25­–38.
I ♥ tori.
MATHEMATICS
IS ALL AROUND US.
236
Any thoughts on this article? Send an
email to [email protected].—Ed.
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
●
Vol. 21, No. 4, November 2015
Rebecca Ambrose,
rcambrose@ucdavis
.edu, works as a teacher
educator at the University
of California–Davis. Her
work focuses on student
explanations and how
teachers can use their
students’ ideas as a
focus of instruction.
Erica Burnison, eburnison.eb@gmail
.com, a math specialist in the Davis Joint
Unified School District, taught middle
school math for sixteen years. She is
a Master Teacher for the BetterLesson
Master Teacher Project and is interested
in making student thinking visible, so that
it can be central to the learning process.
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